%   ORIGINAL: 'h4/thm/list/NULL_FILTER_'
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Goal: !P ls. 'h4/const/list/NULL' ('h4/const/list/FILTER' P ls) <=> (!x. 'h4/const/bool/IN' x ('h4/const/list/LIST_TO_SET' ls) ==> ~P x)
%   PROCESSED
% Assm ['HL_TRUTH']: T
% Assm ['HL_FALSITY']: ~F
% Assm ['HL_BOOL_CASES']: !t. (t <=> T) \/ (t <=> F)
% Assm ['HL_EXT']: !f g. (!x. happ f x = happ g x) ==> f = g
% Goal: !P ls. 'h4/const/list/NULL' ('h4/const/list/FILTER' P ls) <=> (!x. 'h4/const/bool/IN' x ('h4/const/list/LIST_TO_SET' ls) ==> ~happ P x)
fof('HL_TRUTH', axiom, p(s(bool,'T'))).
fof('HL_FALSITY', axiom, ~ (p(s(bool,'F')))).
fof('HL_BOOL_CASES', axiom, ![T]: (s(bool,T) = s(bool,'T') | s(bool,T) = s(bool,'F'))).
fof('HL_EXT', axiom, ![V_3f5439,V_3f5435]: ![F, G]: (![X]: s(V_3f5435,happ(s(fun(V_3f5439,V_3f5435),F),s(V_3f5439,X))) = s(V_3f5435,happ(s(fun(V_3f5439,V_3f5435),G),s(V_3f5439,X))) => s(fun(V_3f5439,V_3f5435),F) = s(fun(V_3f5439,V_3f5435),G))).
fof('h4/thm/list/NULL_FILTER_', conjecture, ![A]: ![P, Ls]: (p(s(bool,'h4/const/list/NULL'(s('h4/type/list/list'(A),'h4/const/list/FILTER'(s(fun(A,bool),P),s('h4/type/list/list'(A),Ls)))))) <=> ![X]: (p(s(bool,'h4/const/bool/IN'(s(A,X),s(fun(A,bool),'h4/const/list/LIST_TO_SET'(s('h4/type/list/list'(A),Ls)))))) => ~ (p(s(bool,happ(s(fun(A,bool),P),s(A,X)))))))).
